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In this paper we consider the following optimal stopping problem V A ω (s) = sup τ ∈ T E s [ e - ∫ 0 τ ω (S w) d w g (S τ) ] , where the process S t is a jump-diffusion process, T is a family of stopping times while g and ω are fixed payoff function and discount function, respectively. In a financial market context, if g (s) = (K - s) + or g (s) = (s - K) + and E is the expectation taken with respect to a martingale measure, V A ω (s) describes the price of a perpetual American option with a discount rate depending on the value of the asset process S t . If ω is a constant, the above problem produces the standard case of pricing perpetual American options. In the first part of this paper we find sufficient conditions for the convexity of the value function V A ω (s) . This allows us to determine the stopping region as a certain interval and hence we are able to identify the form of V A ω (s) . We also present a put-call symmetry for American options with asset-dependent discounting. In the case when S t is a spectrally negative geometric Lévy process we give exact expressions using the so-called omega scale functions introduced in [30]. We show that the analysed value function satisfies the HJB equation and we give sufficient conditions for the smooth fit property as well. Finally, we present a few examples for which we obtain the analytical form of the value function V A ω (s) . [ABSTRACT FROM AUTHOR]

We study critical GI/G/1 queues under finite second-moment assumptions. We show that the busy-period distribution is regularly varying with index half. We also review previously known M/G/1/ and M/M/1 derivations, yielding exact asymptotics as well as a similar derivation for GI/M/1. The busy-period asymptotics determine the growth rate of moments of the renewal process counting busy cycles. We further use this to demonstrate a Balancing Reduces Asymptotic Variance of Outputs (BRAVO) phenomenon for the work-output process (namely the busy time). This yields new insight on the BRAVO effect. A second contribution of the paper is in settling previous conjectured results about GI/G/1 and GI/G/s BRAVO. Previously, infinite buffer BRAVO was generally only settled under fourth-moment assumptions together with an assumption about the tail of the busy period. In the current paper, we strengthen the previous results by reducing to assumptions to existence of 2 + ϵ moments. [ABSTRACT FROM AUTHOR]

In this paper we give a few expressions and asymptotics of ruin probabilities for a Markov modulated risk process for various regimes of a time horizon, initial reserves and a claim size distribution. We also consider a few versions of the ruin time. [ABSTRACT FROM AUTHOR]

In this paper we develop the Gerber-Shiu theory for the classic and dual discrete risk processes in a Markovian (regime switching) environment. In particular, by expressing the Gerber-Shiu function in terms of potential measures of an upward (downward) skip-free discrete-time and discrete-space Markov Additive Process (MAP), we derive closed form expressions for the Gerber-Shiu function in terms of the so-called (discrete) W v and Z v scale matrices, which were introduced in [27]. We show that the discrete scale matrices allow for a unified approach for identifying the Gerber-Shiu function as well as the value function of the associated constant dividend barrier problems. [ABSTRACT FROM AUTHOR]

To get exponential asymptotics for general distributions of HT ht , HT ht (for example geometric ones), one can follow [[3], [6], [12]]. At the beginning of the I n i th time slot, HT ht customers arrive to both queues. Let HT ht with HT ht be the queue length after the service HT ht and before the arrival HT ht . [Extracted from the article]

In this note, we prove that the speed of convergence of the workload of a Lévy-driven queue to the quasi-stationary distribution is of order 1/t. We identify also the Laplace transform of the measure giving this speed and provide some examples. [ABSTRACT FROM AUTHOR]

In this paper, we analyse some equity-linked contracts that are related to drawdown and drawup events based on assets governed by a geometric spectrally negative Lévy process. Drawdown and drawup refer to the differences between the historical maximum and minimum of the asset price and its current value, respectively. We consider four contracts. In the first contract, a protection buyer pays a premium with a constant intensity p until the drawdown of fixed size occurs. In return, he/she receives a certain insured amount at the drawdown epoch, which depends on the drawdown level at that moment. Next, the insurance contract may expire earlier if a certain fixed drawup event occurs prior to the fixed drawdown. The last two contracts are extensions of the previous ones but with an additional cancellable feature that allows the investor to terminate the contracts earlier. In these cases, a fee for early stopping depends on the drawdown level at the stopping epoch. In this work, we focus on two problems: calculating the fair premium p for basic contracts and finding the optimal stopping rule for the polices with a cancellable feature. To do this, we use a fluctuation theory of Lévy processes and rely on a theory of optimal stopping. [ABSTRACT FROM AUTHOR]

We consider in this paper a risk reserve process where the claims and gains arrive according to two independent Poisson processes. While the gain sizes are phase-type distributed, we assume instead that the claim sizes are phase-type perturbed by a heavy-tailed component; that is, the claim size distribution is formally chosen to be phase-type with large probability 1 − ϵ and heavy-tailed with small probability ϵ. We analyze the seminal Gerber-Shiu function coding the joint distribution of the time to ruin, the surplus immediately before ruin, and the deficit at ruin. We derive its value as an expansion with respect to powers of ϵ with known coefficients and we construct approximations from the first two terms of the aforementioned series. The main idea is based on the so-called fluid embedding that allows to put the considered risk process into the framework of spectrally negative Markov-additive processes and use its fluctuation theory developed in Ivanovs and Palmowski 2012. [ABSTRACT FROM AUTHOR]

In this paper, we give solution to the quickest drift change detection problem for a Lévy process consisting of both a continuous Gaussian part and a jump component. We consider here Bayesian framework with an exponential a priori distribution of the change point using an optimality criterion based on a probability of false alarm and an expected delay of the detection. Our approach is based on the optimal stopping theory and solving some boundary value problem. Paper is supplemented by an extensive numerical analysis related with the construction of the Generalized Shiryaev-Roberts statistics. In particular, we apply this method (after appropriate calibration) to analyse Polish life tables and to model the force of mortality in this population with a drift changing in time. [ABSTRACT FROM AUTHOR]

This paper concerns the dual risk model, dual to the risk model for insurance applications, where premiums are surplus-dependent. In such a model, premiums are regarded as costs and claims refer to profits. We calculate the mean of the cumulative discounted dividends paid until the time of ruin, if the barrier strategy is applied. We formulate the associated Hamilton-Jacobi-Bellman equation and identify sufficient conditions for a barrier strategy to be optimal. Numerical examples are provided. [ABSTRACT FROM AUTHOR]

Abstract In this paper we solve the exit problems for (reflected) spectrally negative Lévy processes, which are exponentially killed with a killing intensity dependent on the present state of the process and analyze respective resolvents. All identities are given in terms of new generalizations of scale functions. For the particular cases ω (x) = q and ω (x) = q 1 (a , b) (x) , we obtain results for the classical exit problems and the Laplace transforms of the occupation times in a given interval, until first passage times, respectively. Our results can also be applied to find the bankruptcy probability in the so-called Omega model, where bankruptcy occurs at rate ω (x) when the Lévy surplus process is at level x < 0. Finally, we apply these results to obtain some exit identities for spectrally positive self-similar Markov processes. The main method throughout all the proofs relies on the classical fluctuation identities for Lévy processes, the Markov property and some basic properties of a Poisson process. [ABSTRACT FROM AUTHOR]

In this article, we provide predictable and chaotic representations for Itô-Markov additive processes X. Such a process is governed by a finite-state continuous time Markov chain J which allows one to modify the parameters of the Itô-jump process (in so-called regime switching manner). In addition, the transition of J triggers the jump of X distributed depending on the states of J just prior to the transition. This family of processes includes Markov modulated Itô-Lévy processes and Markov additive processes. The derived chaotic representation of a square-integrable random variable is given as a sum of stochastic integrals with respect to some explicitly constructed orthogonal martingales. We identify the predictable representation of a square-integrable martingale as a sum of stochastic integrals of predictable processes with respect to Brownian motion and power-jumps martingales related to all the jumps appearing in the model. This result generalizes the seminal result of Jacod-Yor and is of importance in financial mathematics. The derived representation then allows one to enlarge the incomplete market by a series of power-jump assets and to price all market-derivatives. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of maximizing the expected discounted utility of dividend payments for an insurance company that controls risk exposure by purchasing proportional reinsurance. We assume the preference of the insurer is of CRRA form. By solving the corresponding Hamilton-Jacobi-Bellman equation, we identify the value function and the corresponding optimal strategy. We also analyze the asymptotic behavior of the value function for large initial reserves. Finally, we provide some numerical examples to illustrate the results and analyze the sensitivity of the parameters. [ABSTRACT FROM AUTHOR]

In this paper we evaluate the probability of the discrete time Parisian ruin that occurs when surplus process stays below or at zero at least for some fixed duration of time d > 0. We identify expressions for the ruin probabilities within finite and infinite-time horizon. We also find their light and heavy-tailed asymptoticswhen initial reserves approach infinity. Finally, we calculate these probabilities for a few explicit examples. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a sub-class of two-dimensional homogeneous nearest neighbor (simple) random walk restricted on the lattice using the matrix geometric approach. In particular, we first present an alternative approach for the calculation of the stability condition, extending the result of Neuts drift conditions[30] and connecting it with the result of Fayolle et al. which is based on Lyapunov functions.[13] Furthermore, we consider the sub-class of random walks with equilibrium distributions given as series of product forms and, for this class of random walks, we calculate the eigenvalues and the corresponding eigenvectors of the infinite matrix R appearing in the matrix geometric approach. This result is obtained by connecting and extending three existing approaches available for such an analysis: the matrix geometric approach, the compensation approach and the boundary value problem method. In this paper, we also present the spectral properties of the infinite matrix R. [ABSTRACT FROM AUTHOR]

In this paper we consider the insurance polices based on drawdown and drawup events where an underlying asset is derived by a classical risk process with phasetype claim sizes perturbed by Brownian motion. The drawdown/drawup process we define as a difference between the historical maximum/minimum and current asset value. We consider four contracts presented in [Palmowski, Tumilewicz 2016]. The first one is an insurance contract where the protection buyer is paying a constant premium with intensity p until the drawdown of fixed size occurs. In return he/she receives a certain insured amount at the drawdown epoch. The second insurance contract may expire early if a certain fixed drawup event occurs prior to a fixed drawdown. The last two contracts are extensions of the previous ones by an additional cancellable feature which allows an investor to terminate the contract earlier. We focus here on an extensive numerical analysis when claim sizes are phase-type. [ABSTRACT FROM AUTHOR]

Nowadays the insurance industry is facing huge challenges related to longevity risk, i.e. the risk that the trend of longevity growth significantly changes in the future. One of the crucial steps in dealing with it is identifying the change of the mortality rate drift observed in prospective life tables. The purpose of this article is to identify this change by casting the problem of quickest detection in the framework of optimal stopping theory. We construct generalized discrete-time Shiryaev's-Roberts statistics and we use it in the analysis of Polish life tables from years 1990-2014. We model the logarithm of the mortal intensity by the Brownian motion that changes the zero drift into nonzero one at some random time. For this case we construct optimal stopping rule detecting substantial change of this drift. We also present calibration of above model using Central Statistical Office data and we carry out extensive statistical analysis showing huge potential of described statistics. [ABSTRACT FROM AUTHOR]

This paper concerns an optimal dividend distribution problem for an insurance company with surplus-dependent premium. In the absence of dividend payments, such a risk process is a particular case of so-called piecewise deterministic Markov processes. The control mechanism chooses the size of dividend payments. The objective consists in maximizing the sum of the expected cumulative discounted dividend payments received until the time of ruin and a penalty payment at the time of ruin, which is an increasing function of the size of the shortfall at ruin. A complete solution is presented to the corresponding stochastic control problem. We identify the associated Hamilton-Jacobi-Bellman equation and find necessary and sufficient conditions for optimality of a single dividend-band strategy, in terms of particular Gerber-Shiu functions. A number of concrete examples are analyzed. [ABSTRACT FROM AUTHOR]

In this paper, we consider dividend problem for an insurance company whose risk evolves as a spectrally negative Lévy process (in the absence of dividend payments) when a Parisian delay is applied. An objective function is given by the cumulative discounted dividends received until the moment of ruin, when a so-called barrier strategy is applied. Additionally, we consider two possibilities of a delay. In the first scenario, ruin happens when the surplus process stays below zero longer than a fixed amount of time. In the second case, there is a time lag between the decision of paying dividends and its implementation. [ABSTRACT FROM AUTHOR]

We consider a controlled queuing model with Lévy input. The controls take place at random times. They involve the current workload and the input processes and may also depend on whether the workload process has reached certain critical values since the last control epoch. We propose a solution strategy for deriving the steady-state distribution of this model which is based on recent advances in the fluctuation theory of spectrally one-sided Lévy process. We provide illustrative examples involving a clearing model, an inventory model, and a model for the TCP protocol. [ABSTRACT FROM AUTHOR]

Abstract: This paper solves exit problems for spectrally negative Markov additive processes and their reflections. So-called scale matrix, which is a generalization of the scale function of a spectrally negative Lévy process, plays the central role in the study of the exit problems. Existence of the scale matrix was shown by Kyprianou and Palmowski (2008) . We provide the probabilistic construction of the scale matrix, and identify its transform. In addition, we generalize to the MAP setting the relation between the scale function and the excursion (height) measure. The main technique is based on the occupation density formula and even in the context of fluctuations of spectrally negative Lévy processes this idea seems to be new. Our representation of the scale matrix in terms of nice probabilistic objects opens up possibilities for further investigation of its properties. [Copyright &y& Elsevier]

This article we analyzes the quasi-stationary workload of a Lévy-driven storage system. More precisely, assuming the system is in stationarity, we study its behavior conditional on the event that the busy period T in which time 0 is contained has not ended before time t, as t → ∞. We do so by first identifying the double Laplace transform associated with the workloads at time 0 and time t, on the event {T > t}. This transform can be explicitly computed for the case of spectrally one-sided jumps. Then asymptotic techniques for Laplace inversion are relied upon to find the corresponding behavior in the limiting regime that t → ∞. Several examples are treated; for instance in the case of Brownian input, we conclude that the workload distribution at time 0 and t are both Erlang(2). [ABSTRACT FROM AUTHOR]

The areas under the workload process and under the queueing process in a single-server queue over the busy period have many applications not only in queueing theory but also in risk theory or percolation theory. We focus here on the tail behaviour of distribution of these two integrals. We present various open problems and conjectures, which are supported by partial results for some special cases. [ABSTRACT FROM AUTHOR]

Abstract: We consider a queuing model with the workload evolving between consecutive i.i.d. exponential timers according to a spectrally positive Lévy process which is reflected at . When the exponential clock ends, the additional state-dependent service requirement modifies the workload so that the latter is equal to at epoch for some random nonnegative i.i.d. functionals . In particular, we focus on the case when , where are i.i.d. nonnegative random variables. We analyse the steady-state workload distribution for this model. [Copyright &y& Elsevier]

Consider two insurance companies (or two branches of the same company) that receive premiums at different rates and then split the amount they pay in fixed proportions for each claim (for simplicity we assume that they are equal). We model the occurrence of claims according to a Poisson process. The ruin is achieved when the corresponding two-dimensional risk process first leaves the positive quadrant. We will consider two scenarios of the controlled process: refraction and impulse control. In the first case the dividends are payed out when the two-dimensional risk process exits the fixed region. In the second scenario, whenever the process hits the horizontal line, it is reduced by paying dividends to some fixed point in the positive quadrant where it waits for the next claim to arrive. In both models we calculate the discounted cumulative dividend payments until the ruin. This article is the first attempt to understand the effect of dependencies of two portfolios on the joint optimal strategy of paying dividends. For example in case of proportional reinsurance one can observe the interesting phenomenon that choice of the optimal barrier depends on the initial reserves. This is in contrast with the one-dimensional Cramer-Lundberg model where the optimal choice of the barrier is uniform for all initial reserves. [ABSTRACT FROM AUTHOR]

Stochastic fluid-fluid models (SFFMs) offer powerful modeling ability for a wide range of real-life systems of significance. The existing theoretical framework for this class of models is in terms of operator-analytic methods. For the first time, we establish matrix-analytic methods for the efficient analysis of SFFMs. We illustrate the theory with numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, I analyze the distributional properties of the busy period in an on-off fluid queue and the first passage time in a fluid queue driven by a finite state Markov process. In particular, I show that the first passage time has a IFR distribution and the busy period in the Anick-Mitra-Sondhi model has a DFR distribution. [ABSTRACT FROM AUTHOR]

Abstract.Quasi-birth-and-death processes (QBDs) form a class of Markovian models with many applications in which the level-variable X(t) records the number of individuals in some system and a phase-variable φ(t) records some additional information about the system. We study the sensitivity of various performance measures of QBDs with respect to the perturbation of a generator matrix. Thereby, we extend the sensitivity analysis of level-dependent QBDs in Gomez-Corral and Lopez-Garcia[5] and develop results with a focus on applications in modelling healthcare systems. [ABSTRACT FROM AUTHOR]

In this paper we propose new iterative algorithm of calculating the joint distribution of the Parisian ruin time and the number of claims until Parisian ruin for the classical risk model. Examples are provided when the generic claim size is mixed Erlang distributed. We focus on the exponentially and the Erlang ( k , μ ) ( k ≥ 2 ) distributed claim sizes. [ABSTRACT FROM AUTHOR]

We study the all-time supremum of the perturbed branching random walk, known to be the endogenous solution to the high-order Lindley equation: W = D max Y , max 1 ≤ i ≤ N (W i + X i) , where the { W i } are independent copies of W , independent of the random vector (Y , N , { X i }) taking values in R × N × R ∞ . Under Kesten assumptions, this solution satisfies P (W > t) ∼ H e − α t , t → ∞ , where α > 0 solves the Cramér–Lundberg equation E ∑ i = 1 N e α X i = 1. This paper establishes the tail asymptotics of W by using the forward iterations of the map defining the fixed-point equation combined with a change of measure along a randomly chosen path. This new approach provides an explicit representation of the constant H and gives rise to unbiased and strongly efficient estimators for the rare event probabilities P (W > t). [ABSTRACT FROM AUTHOR]

In this article, we analyze a branching process with immigration defined recursively byXt = θt○Xt−1 + Btfor a sequence (Bt) of i.i.d. random variables and random mappings, withbeing a sequence of ℕ0-valued i.i.d. random variables independent ofBt. We assume that one of generic variablesAandBhas a regularly varying tail distribution. We identify the tail behavior of the distribution of the stationary solutionXt. We also prove CLT for the partial sums that could be further generalized to FCLT. Finally, we also show that partial maxima have a Fréchet limiting distribution. [ABSTRACT FROM AUTHOR]

In this paper we develop a symbolic technique to obtain asymptotic expressions for ruin probabilities and discounted penalty functions in renewal insurance risk models when the premium income depends on the present surplus of the insurance portfolio. The analysis is based on boundary problems for linear ordinary differential equations with variable coefficients. The algebraic structure of the Green's operators allows us to develop an intuitive way of tackling the asymptotic behavior of the solutions, leading to exponential-type expansions and Cramér-type asymptotics. Furthermore, we obtain closed-form solutions for more specific cases of premium functions in the compound Poisson risk model. [ABSTRACT FROM AUTHOR]

This paper presents an analysis of the stochastic recursion W i + 1 = [ V i W i + Y i ] + that can be interpreted as an autoregressive process of order 1, reflected at 0. We start our exposition by a discussion of the model's stability condition. Writing Y i = B i - A i , for independent sequences of nonnegative i.i.d. random variables { A i } i ∈ N 0 and { B i } i ∈ N 0 , and assuming { V i } i ∈ N 0 is an i.i.d. sequence as well (independent of { A i } i ∈ N 0 and { B i } i ∈ N 0 ), we then consider three special cases (i) V i equals a positive value a with certain probability p ∈ (0 , 1) and is negative otherwise, and both A i and B i have a rational LST, (ii) V i attains negative values only and B i has a rational LST, (iii) V i is uniformly distributed on [0, 1], and A i is exponentially distributed. In all three cases, we derive transient and stationary results, where the transient results are in terms of the transform at a geometrically distributed epoch. [ABSTRACT FROM AUTHOR]